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Fitting

- We’ve learned how to detect edges, corners, blobs. Now what?
- We would like to form a higher-level, more compact representation of the features in the image by grouping multiple features according to a simple model

9300 Harris Corners Pkwy, Charlotte, NC

Fitting

- Choose a parametric model to represent a set of features

simple model: circles

simple model: lines

complicated model: car

Source: K. Grauman

Fitting: Issues

- Noise in the measured feature locations
- Extraneous data: clutter (outliers), multiple lines
- Missing data: occlusions

Case study: Line detection

Fitting: Overview

- If we know which points belong to the line, how do we find the “optimal” line parameters?
- Least squares
- What if there are outliers?
- Robust fitting, RANSAC
- What if there are many lines?
- Voting methods: RANSAC, Hough transform
- What if we’re not even sure it’s a line?
- Model selection

Least squares line fitting

- Data: (x1, y1), …, (xn, yn)
- Line equation: yi = mxi + b
- Find (m, b) to minimize

y=mx+b

(xi, yi)

Normal equations: least squares solution to XB=Y

Problem with “vertical” least squares

- Not rotation-invariant
- Fails completely for vertical lines

Total least squares

- Distance between point (xi, yi) and lineax+by=d (a2+b2=1): |axi + byi – d|

ax+by=d

Unit normal: N=(a, b)

(xi, yi)

Total least squares

- Distance between point (xi, yi) and lineax+by=d (a2+b2=1): |axi + byi – d|
- Find (a, b, d) to minimize the sum of squared perpendicular distances

ax+by=d

Unit normal: N=(a, b)

(xi, yi)

Total least squares

- Distance between point (xi, yi) and lineax+by=d (a2+b2=1): |axi + byi – d|
- Find (a, b, d) to minimize the sum of squared perpendicular distances

ax+by=d

Unit normal: N=(a, b)

(xi, yi)

Solution to (UTU)N = 0, subject to ||N||2 = 1: eigenvector of UTUassociated with the smallest eigenvalue (least squares solution to homogeneous linear systemUN = 0)

Total least squares

second moment matrix

Least squares as likelihood maximization

- Generative model: line points are sampled independently and corrupted by Gaussian noise in the direction perpendicular to the line

ax+by=d

(u, v)

ε

(x, y)

point on the line

normaldirection

noise:sampled from zero-meanGaussian withstd. dev. σ

Least squares as likelihood maximization

- Generative model: line points are sampled independently and corrupted by Gaussian noise in the direction perpendicular to the line

ax+by=d

(u, v)

ε

(x, y)

Likelihood of points given line parameters (a, b, d):

Least squares: Robustness to noise

- Least squares fit to the red points:

Least squares: Robustness to noise

- Least squares fit with an outlier:

Problem: squared error heavily penalizes outliers

Robust estimators

- General approach: find model parameters θ that minimizeri(xi, θ) – residual of ith point w.r.t. model parametersθρ – robust function with scale parameter σ

The robust function ρ behaves like squared distance for small values of the residual u but saturates for larger values of u

Choosing the scale: Just right

The effect of the outlier is minimized

Choosing the scale: Too small

The error value is almost the same for everypoint and the fit is very poor

Choosing the scale: Too large

Behaves much the same as least squares

Robust estimation: Details

- Robust fitting is a nonlinear optimization problem that must be solved iteratively
- Least squares solution can be used for initialization
- Adaptive choice of scale: approx. 1.5 times median residual (F&P, Sec. 15.5.1)

RANSAC

- Robust fitting can deal with a few outliers – what if we have very many?
- Random sample consensus (RANSAC): Very general framework for model fitting in the presence of outliers
- Outline
- Choose a small subset of points uniformly at random
- Fit a model to that subset
- Find all remaining points that are “close” to the model and reject the rest as outliers
- Do this many times and choose the best model

M. A. Fischler, R. C. Bolles. Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography. Comm. of the ACM, Vol 24, pp 381-395, 1981.

RANSAC for line fitting example

Source: R. Raguram

RANSAC for line fitting example

- Randomly select minimal subset of points
- Hypothesize a model

Source: R. Raguram

RANSAC for line fitting example

- Randomly select minimal subset of points
- Hypothesize a model
- Compute error function

Source: R. Raguram

RANSAC for line fitting example

- Randomly select minimal subset of points
- Hypothesize a model
- Compute error function
- Select points consistent with model

Source: R. Raguram

RANSAC for line fitting example

- Randomly select minimal subset of points
- Hypothesize a model
- Compute error function
- Select points consistent with model
- Repeat hypothesize-and-verify loop

Source: R. Raguram

RANSAC for line fitting example

- Randomly select minimal subset of points
- Hypothesize a model
- Compute error function
- Select points consistent with model
- Repeat hypothesize-and-verify loop

Source: R. Raguram

RANSAC for line fitting example

Uncontaminated sample

- Randomly select minimal subset of points
- Hypothesize a model
- Compute error function
- Select points consistent with model
- Repeat hypothesize-and-verify loop

Source: R. Raguram

RANSAC for line fitting example

- Randomly select minimal subset of points
- Hypothesize a model
- Compute error function
- Select points consistent with model
- Repeat hypothesize-and-verify loop

Source: R. Raguram

RANSAC for line fitting

- Repeat N times:
- Draw s points uniformly at random
- Fit line to these s points
- Find inliers to this line among the remaining points (i.e., points whose distance from the line is less than t)
- If there are d or more inliers, accept the line and refit using all inliers

Choosing the parameters

- Initial number of points s
- Typically minimum number needed to fit the model
- Distance threshold t
- Choose t so probability for inlier is p (e.g. 0.95)
- Zero-mean Gaussian noise with std. dev. σ: t2=3.84σ2
- Number of samples N
- Choose N so that, with probability p, at least one random sample is free from outliers (e.g. p=0.99) (outlier ratio: e)

Source: M. Pollefeys

Choosing the parameters

- Initial number of points s
- Typically minimum number needed to fit the model
- Distance threshold t
- Choose t so probability for inlier is p (e.g. 0.95)
- Zero-mean Gaussian noise with std. dev. σ: t2=3.84σ2
- Number of samples N
- Choose N so that, with probability p, at least one random sample is free from outliers (e.g. p=0.99) (outlier ratio: e)

Source: M. Pollefeys

Choosing the parameters

- Initial number of points s
- Typically minimum number needed to fit the model
- Distance threshold t
- Choose t so probability for inlier is p (e.g. 0.95)
- Zero-mean Gaussian noise with std. dev. σ: t2=3.84σ2
- Number of samples N
- Choose N so that, with probability p, at least one random sample is free from outliers (e.g. p=0.99) (outlier ratio: e)

Source: M. Pollefeys

Choosing the parameters

- Initial number of points s
- Typically minimum number needed to fit the model
- Distance threshold t
- Choose t so probability for inlier is p (e.g. 0.95)
- Zero-mean Gaussian noise with std. dev. σ: t2=3.84σ2
- Number of samples N
- Consensus set size d
- Should match expected inlier ratio

Source: M. Pollefeys

Adaptively determining the number of samples

- Inlier ratio e is often unknown a priori, so pick worst case, e.g. 50%, and adapt if more inliers are found, e.g. 80% would yield e=0.2
- Adaptive procedure:
- N=∞, sample_count =0
- While N >sample_count
- Choose a sample and count the number of inliers
- Set e = 1 – (number of inliers)/(total number of points)
- Recompute N from e:
- Increment the sample_count by 1

Source: M. Pollefeys

RANSAC pros and cons

- Pros
- Simple and general
- Applicable to many different problems
- Often works well in practice
- Cons
- Lots of parameters to tune
- Doesn’t work well for low inlier ratios (too many iterations, or can fail completely)
- Can’t always get a good initialization of the model based on the minimum number of samples

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